This may be a silly question, but I've never seen an unripe vegetable. Is the growing of a vegetable different from a fruit's? I could pick a carrot up from the dirt whenever, and eat it, but I would never do that to a strawberry.
Also, is it dangerous to eat particular vegetables (potatoes, I believe) before they're fully grown?
Yes, the other answers are right, there is no such ripening. I am writing this answer to explain in more depth why this is so.
From a plant's point of view, a botanical fruit (b.fruit) is the package for its seeds. Its purpose is to be eaten by animals, so that the seeds can germinate somewhere far away, avoiding resource competition with its mother and sister plants. To entice animals to eat the b.fruits, the plants have evolved b.fruits which become easy to spot and tasty (= full of energy-dense and easily digested sugars) at the same time as the seeds are mature enough for reproduction. The plant expends quite a lot of its limited energy supply on creating nutrients for its seeds and sugars, pigments and fruity-smelling esters for its b.fruit, and this process takes time. When it is finalized, the b.fruit is what we call "ripe".
This process works the same way for any plant with typical b.fruits, including the ones we eat as culinary fruits (strawberries, apples), the ones we eat as vegetables (tomatoes, aubergines) and the ones we don't eat (yew berries). There are of course exceptions, for example maple seeds get dispersed by the wind, and don't get "plumped" to be tasty for animals.
Of course, the plant can choose to expend its energy (which is the generic currency for life) for other purposes. Instead of adding mass to its b.fruit, it can add mass to any other part of its body - stems, leaves, etc. It can also create intricate nonbulky molecules for these body parts. Some plants can even chemically save energy in body parts which function as "savings accounts" such as tubers (potatoes) or modified roots (carrots, parsnips, onions).
But from the plant's point of view, these body parts are not meant to be eaten, unlike b.fruits. So, when the plant invests energy in them, it does it in ways which support the body part's role. Especially for structural parts like stems, this makes them stronger, so tougher and woodier. But also other parts like leaves or roots tend to get structurally reinforced with time. Others don't change much, except to increase in size. If the plant uses unpleasant molecules to deter eating, it also needs time to build up a supply. There certainly is no swift transition to an animal-enticing state, just slow growth, and in some cases, toughening or a slow buildup of bitter or otherwise unpleasant taste.
This is why it is culinary optimal to pick vegetables (the ones which are not b.fruits) when they are young, with a few exceptions (e.g. for dolmas, you want both the larger area and the stronger taste of older grape leaves). This goes against the economic interest of the grower, because the tastier young plants have less mass.
In the end, you can eat any fruit or vegetable at any stage, but it is unlikely that you would enjoy the taste if you veer too far off from the traditional picking stage.
Vegetables don't ripen. Carrots, potatoes, leafy greens, etc...
They will get bigger of course and some veggies will get more bitter or woody with age. Greens will get bitter.
Generally speaking veggies are better young. Carrots are particularly sweet and tender when young. Potatoes are also perfectly fine at any size.
The possible exceptions would be the fruits that are labelled vegetables for cooking purposes like some squashes and tomatoes. These do need to be ripe and whether they can be ripened after picking will depend on the plant.
Surprisingly, this is more of a gardening question than a cooking question. While ripening isn't the word you'd use (that applies pretty strictly to fruit), some vegetables, especially greens, go through a process called bolting, which is basically the intermediate stage between edible greens and going to seed. Usually you don't want to eat what results. Lettuce is particularly bad about this in hot weather -- once the stem starts developing, the leaves turn sort of fern-like and very bitter, and the plant fills with latex. Needless to say, unless you're trying to torture a head of roots-on hydro lettuce, this will seldom be a concern in the kitchen.
Fruits are parts of plants that contain seeds (apples, tomatoes, cucumbers, peepers, okra, lemons). What people consider ripe fruit is not always related to the botanical ripeness (the seeds are mature and ready to be dispersed)
Vegetables are all the other parts of plants that we eat
Vegetables do not ripen, they just grow. Many plants are better tasting and more edible at certain times in their growth cycle, but this is generally based on your culture and preference, there are no firm rules. Very few plants change from being poisonous to be edible during their growth cycle
All fruits and vegetables begin to decay the day they are harvested. In some fruits this is to their taste advantage, in most vegetables this is not. Fruits that are picked before they are ripe may improve in flavour and texture over a few days of suitable storage
In most cases fruit and vegetable nutritional profiles decay from the day they are harvested
